Question

Lim (1 - 1/(n + 1)) ^ (n ^ 2)

Answer 1

The limit of (1 - 1/(n + 1)) ^ (n ^ 2) as n approaches infinity is 0.

As n becomes larger, the term 1/(n + 1) becomes smaller and closer to zero, which means that (1 - 1/(n + 1)) becomes closer and closer to 1. Therefore, the expression (1 - 1/(n + 1)) ^ (n ^ 2) becomes closer and closer to 1^(n^2) = 1. And any number powered by infinity (n^2) will be approaching zero, so the limit of the expression is 0.

Answer 2

Answer: Lim (1 - 1/(n + 1)) ^ (n ^ 2) = 1^infinity = 1

Step-by-step explanation: The expression given is the limit of the sequence (1 - 1/(n + 1)) ^ (n ^ 2) as n approaches infinity.

We can start by looking at the exponent first. The n^2 will grow faster than any polynomial function, making the entire expression go to zero as n approaches infinity.

Now let's look at the base (1 - 1/(n + 1))

As n increases, the value of the base becomes closer and closer to 1, since 1/(n+1) becomes smaller and smaller.

Therefore, the limit of the expression is:

Lim (1 - 1/(n + 1)) ^ (n ^ 2) = 1^infinity = 1

As n goes to infinity, the expression goes to 1

Please note that this is true for the limit, for a specific value of n, the expression will not be 1.